Calculation fluorites

This paper presents the results of ab initio calculation investigating the pressure dependence of properties of rutile, anatase and brookite, as well as of columbite and hypothetical fluorite phases. The main emphasis is on lattice properties since it was possible to locate transitions and investigate transformation precursors by using constant-pressure optimization algorithm. 

The fluorite phase is found to be extremely high in energy (it falls outside the energy range of Figure 1). Its equilibrium volume at P=0 would be 27.648 A /mol, and calculated equation of state gives B=287 GPa and B"=4.18. These values make fluorite structure the least compressible of all titanium dioxide polymorphs studied here, but still leaves the observation of a phase with B>500 GPa unexplained. ... 

Due to the intermediate coupling the sign of the crystal field matrix element 6 is reversed compared to the pure Russell-Saunders state. Thus for 8-fold cubic coordination a F7 ground state was found. From EPR measurements on Pu3"1" diluted in fluorite host lattices, a magnetic moment at T=0 K can be calculated, ranging from li ff = 1.333 (in Ce02) to y ff = 0.942 (in SrCl2) (24,... 

Calculate the number of cations, anions, and formula units per unit cell in each of the following solids (a) the rock-salt unit cell shown in Fig. 5.39 (h) the fluorite (CaF2) unit cell shown here, (c) What are the coordination numbers of the ions in fluorite ... 

Powder XR diffraction spectra confirm that all materials are single phase solid solutions with a cubic fluorite structure. Even when 10 mol% of the cations is substituted with dopant the original structure is retained. We used Kim s formula (28) and the corresponding ion radii (29) to estimate the concentration of dopant in the cerium oxide lattice. The calculated lattice parameters show that less dopant is present in the bulk than expected. As no other phases are present in the spectrum, we expect dopant-enriched crystal surfaces, and possibly some interstitial dopant cations. However, this kind of surface enrichment cannot be determined by XR diffraction owing to the lower ordering at the surface. 

Amy Berger helped me write Chapter 10 (Surface Complexation), and Chapter 31 (Acid Drainage) is derived in part from her work. Edward Warren and Richard Worden of British Petroleum s Sunbury lab contributed data for calculating scaling in North Sea oil fields, Richard Wendlandt first modeled the effects of alkali floods on clastic reservoirs, and Kenneth Sorbie helped write Chapter 30 (Petroleum Reservoirs). I borrowed from Elisabeth Rowan s study of the genesis of fluorite ores at the Albigeois district, Wendy Harrison s study of the Gippsland basin, and a number of other published studies, as referenced in the text. 

The favored defect type in strontium fluoride, which adopts the fluorite structure, are Frenkel defects on the anion sublattice. The enthalpy of formation of an anion Frenkel defect is estimated to be 167.88 kJ mol-1. Calculate the number of F- interstitials and vacancies due to anion Frenkel defects per cubic meter in SrF2 at 1000°C. The unit cell is cubic, with a cell edge of 0.57996 nm and contains four formula units of SrF2. It is reasonable to assume that the number of suitable interstitial sites is half that of the number of anion sites. 

Catlow and Lidiard calculated, by computer assisted cluster calculations in an ionic model, that the 2 2 2 and 4 3 2 clusters are particularly stable. Similar clusters are reported to exist in other ionic fluorite-structure solids, e.g. Cap2 -I- YF3 , indicating that they are a feature of anion-excess fluorite compounds. 

Calculate the heat of formation of fluorite, given the following information (units are kJ mol-1) ... 

Tt is found empirically, indeed, that these complicated and unreliable calculations need not be made in general even for substances of un-symmetrical valence type the interionic distances are very closely approximated by the sum of the crystal radii. For fluorite this sum is 2.35 A, which agrees very well with the observed value. The reason for this is apparent in the crystal radius of Ca++ a correction for bivalence of cation and anion is made, and this has nearly the same magnitude as the correction for bivalence of cation alone made for the sum of the univalent radii of calcium and fluorine. 

To substantiate the need for indicating the crystallographic direction along with hardness test results, we calculate the hardness of one of the fundamental standard blocks of the Mohs scale, i.e., fluorite (H = 4), in three directions... 

Using the density of calcium fluoride (CaF2, found in nature as the mineral fluorite), which is known to be 3.180 g-cm-3, and the information given in Exercise 5.34, calculate (a) the edge length a of the unit cell and (b) the Ca-F separation in fluorite. 

We first look at the fluorides of barium. Only BaF2 is known, a typically ionic solid having the fluorite (8 4) structure. From Table 5.2, we see that the calculated lattice energy is very close to the experimental value in other words, we can calculate the enthalpy of formation of BaF2(s) almost within the limits of experimental uncertainty. Why have BaF3 and BaF not been prepared Presumably they are thermodynamically unstable with respect to other species. In order to verify this supposition, let us estimate the enthalpies of formation AHf of BaF(s) and BaF3(s), assuming these to be ionic. 

The adsorption of sodium oleate on a fluorite (CaF2) IRE has been quantitatively studied more recently (116). Adsoiption proceeded rapidly during the first two hours, but slowed at longer times, taking as much as 30 hours to reach equilibrium. The formation of interfacial calcium dioleate was clearly indicated by the pair of carboxylate bands at 1571 and 1534 cm 1 in an in - situ spectrum of an adsorbed layer formed from sodium oleate at 9 x 10 5 M. Calculation of the adsoiption isotherm from the FT-IR data, at sodium oleate concentrations between 5 x 10 and 5 x 104 M, yielded results in good agreement with values obtained by other methods. 

Figure 6.11 XRD-RSMs with y offset measured for (a) sbtn and (b) fluorite-SBTN thin films and fitted calculated diffraction patterns for the obtained results, (c) and (d).

Pantiledes s values of r. arc given in Table 14-2. He went on to study the relation between these bands and the zincblende bands (Pantiledes, 1975c), as discussed in Chapter 6. In addition, he has carried out the corresponding analysis for the valence bands in the lluorite structure, to obtain universal bands for that system (unpublished). A more recent band calculation for fluorite itself has been made by Albert, Jouain, and Gout (1977). 

The calculation of Z for the cesium chloride structure is also quite direct and is discussed in Problem 14-2. However, the perturbation theory that we have used becomes inaccurate when we go to the rocksalt structure with Z = 2 or 3, or to the fluorite structure. The assumption of weak coupling restricts us to small percentages of. softening. For Z = 2 and = 5.3 from Table 14-2, Eq. (14-10) leads to a softening of 72 percent, and even larger values arc obtained for Z = 3 and for the fluorites. 

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